Freak Waves in Shallow Water

1. Freak Waves in Shallow Water Josh Moser & Chris Wai

2. Rogue waves are being reported more and more in today’s world.

3. Rogue waves are being reported more and more in today’s world.

4. Rogue waves are being reported more and more in today’s world.

5. Rogue waves are being reported more and more in today’s world.

6. , The dispersion relation describes the physics of the waves One definition of a shallow-water wave is that the wavelength is long compared to the depth. Therefore, . This means that the dispersion relation, . The wave speed then calculates as the same as deep-water waves, , so it is also dispersive.

7. The dispersion relation describes the physics of the waves and depends on wavelength, . The spread of the waves is then dependent on the wavelength as shown in k, the wave number. To create a freak wave, it is thought of having the exact dispersion relation in which multiple waves meet up at some time and space, generating a freak wave.

8. The Korteweg-de Vries equation is a nonlinear, partial differential equation that has applications to water waves

9. Small-amplitude waves in shallow water is a statement of weak nonlinearity

10. Appropriate partial derivatives of and

11. Plugging into the linearized KdV equation to find the ordinary differential equation • which is = 0 • The solution of this ODE is

12. Plugging this solution into So that

13. To find , take the Fourier Transform of the initial condition So that

14. Which is particularly useful when considering an ideal situation in which )

15. So now we have an equation that looks like an Airy Integral

16. Now we can make simple substitutions to exploit what is known

17. We know that this is in the form of the Airy Integral using slow time and slow space scales

18. Substitutions can be made to convert it back to real time and space scales for the wavemaker

19. Substitutions can be made to convert it back to real time and space scales for the wavemaker

20. Now consider different initial conditions

21. The initial condition only considers what the axis looks like when at • Namely the initial condition

22. It is not useful to consider when the freak wave is forms when it is at one end of the tank. • So consider the initial condition where is an arbitrary value we can choose • Because we know the speed the wave is travelling, we can choose when we want the freak wave to form and then calculate how far into the tank the freak wave will occur.

24. Applying the new initial condition we have,

25. Then applying the Fourier Transform we have, We know that the delta function has the property such that, In the case above,

26. So evaluating, Now we know the identity of under the initial conditions . So becomes,

27. Using similar substitutions as earlier, we can rewrite this in Airy integral form Then just as before we have

28. Finally in this form which is no different from earlier except that has been replace with Then using similar substitutions to convert back to real time and space

29. So, We know the speed of the wave and can pick a such that the freak wave will occur somewhere reasonable in the tank.

30. Here are some plots to demonstrate what the wave surface should look like when

32. How do we generate these waves? • In the wave tank in the Pritchard laboratory • The wave maker is a vertical paddle that moves backwards and forwards. It is varying in over time . Let us call the position of the wave paddle in , • Without loss of generality, let us assume a wave tank that extends infinitely in one direction. Which are the conserved finite quantities in this case?

33. Mass flux Mass flux Mass flux

34. To consider flux, we must define the direction from the “inside” to the “outside”. Namely, we must parameterize the function and obtain a vector function. We define . And to parameterize, A vector function that is in a normal direction to is then.

35. Then the normal component of velocity is, In general for water waves, the kinematic free surface boundary condition in one space and one time dimension for an air-water interface is, So by the kinematic free surface boundary condition, Where is the velocity potential and so and are the velocity of the water at the position and time .

36. The mass flux “through” the wave paddle would be the height of the fluid at times the velocity of fluid in the direction. Because water cannot go through the paddle, the water’s velocity at the paddle must match the velocity of the paddle at the paddle,

37. So then the flux through the surface of the wave is given by the integral of the change of the wave surface over time, for all of the wave surface. Namely, Now we can equate the mass flux at the paddle with that of the wave surface to get the relation,

38. Referring back to,

40. In conclusion • Lot’s of differential equations • Very mathy • Had a lot of fun

41. Things to do next and new questions to ask • Find numerical solution and try to replicate results • See how the data can be used to predict freak waves that may come into the coast • Find how the sea in real life translates to boundary and initial conditions • Find the set of the conditions that cause freak waves at the shore • Predict and “control” • Save lives

42. http://myarchive.us/richc/2010/Deadlywavekilss2injures14oncruiseshipinM_146AB/ship50footwave2.jpghttp://myarchive.us/richc/2010/Deadlywavekilss2injures14oncruiseshipinM_146AB/ship50footwave2.jpg • http://graphics8.nytimes.com/images/2006/07/11/science/11wave.1.395.jpg • https://www.google.com/search?q=freak+waves&espv=2&tbm=isch&source=lnms&sa=X&ei=8f_9U4S5HrS_sQS514LgCQ&ved=0CAgQ_AUoAw&biw=1280&bih=923#facrc=_&imgdii=_&imgrc=GW7u_JXGbsENlM%253A%3B6HPoEPp_faOGvM%3Bhttp%253A%252F%252Fwww.ipacific.com%252Fforum%252Findex.php%253Faction%253Ddlattach%253Btopic%253D507.0%253Battach%253D408%3Bhttp%253A%252F%252Fwww.ipacific.com%252Fforum%252Findex.php%253Ftopic%253D507.0%3B350%3B250